Improving the Performance of Fuzzy Minimum ... - mecs-press.org

I. J. Computer Network and Information Security, 2018, 6, 16-26 Published Online June 2018 in MECS (http://www.mecs-press.org/) DOI: 10.5815/ijcnis.2018.06.02

Improving the Performance of Fuzzy Minimum Spanning Tree based Routing Process through PNode Fuzzy Multicasting Approach in MANET Soham Bandyopadhyay Dr.B.C.Roy Polytechnic, Durgapur, India E-mail: [email protected]

Sunil Karforma The University of Burdwan, Burdwan, India E-mail: [email protected] Received: 24 February 2018; Accepted: 18 April 2018; Published: 08 June 2018

Abstract—Mobile Ad-hoc Network (MANET) is mostly decentralized and self-adjustable network system. It is significant to optimize the overall network energy utilization and improve packet sending performance by reducing the errors, generated due to different real-life environmental effects. In this paper, considering atmospheric, environmental change and varying distance for topological change, we try to generate the routing cost. By introducing m-minimum (membership value as m) triangular fuzzy number at interval based cost and energy of the network, we try to handle the uncertain environment. Here we generate both fuzzy minimum spanning tree (FMST) for a given n- nodes network and p-node fuzzy multicast minimum spanning tree (pFMMST), in fuzzy interval based format. Applying the fuzzy credibility distribution we modify the network routing cost and energy utilization for both FMST and pFMMST. Comparing the routing cost and residual energy for FMST and pFMMST of MANET, it is concluded that, pFMMST is better FMST based packet routing approach, with minimum routing cost, optimized total energy utilization and best possible technique to reduce the error which is generated due to the deviation of interval of upper and lower limit data in route cost and residual energy. Index Terms—P-nodes fuzzy multicasting, spanning tree, routing, fuzzy credibility distribution, inverse credibility distribution, m-minimum triangular fuzzy number, standard deviation. I. INTRODUCTION A mobile ad-hoc network (MANET) is also known as wireless ad-hoc network. It is basically a continuous selfgenerating, particular infrastructure-less network of mobile devices, connected wirelessly without the support of a centralized administration. According to different researches for transferring data Copyright © 2018 MECS

from one node to another was explained previously, using most efficient multicasting techniques on MANETs. Location-aided routing in MANET was proposed through [3]. Chen et al. [1] analyzed the use of wireless and mobile technology in the education sector. WLAN optimization at the MAC and Network Levels was explained in [4]. Analyzing minimum cost multicast trees in ad hoc networks was explained in [5]. All these multicasting and broadcasting techniques on MANET [6, 22] with different routing algorithms[31] use crisp dataset and generate point estimated result, without considering real- life uncertain constraints. Here the fuzzy system is a most appropriate tool to deal with such uncertainties. Giving the priority to the fuzzy system, different researchers already used it in several mathematical applications. Minimum spanning tree problem by Zhou et al. [10] with fuzzy edge cost as a chance-constrained model was first formulated by [11]. Chang and Lee explained ranking index for fuzzy edge cost of spanning trees. Almeida et al. [13] proposed a genetic algorithm to derive fuzzy minimum spanning tree. But the application area of the fuzzy system in wireless network routing field [8], is not very much extended. So in this paper, we try to incorporate fuzzy logic for reallife decision making to derive the routing cost and total energy utilization for transferring data packets from one digital device to another in MANET. But all these conclusive results do not point estimate. These are interval estimated. Using this fuzzy system we generate minimum spanning tree (Kruskal, Prims) and shortest path (Fuzzy Dijkstra techniques) [14,15].Through shortest path technique different multicast [2] groups of nodes transfer the data packets using less energy utilization as compared to data transfer scenario amongst all the nodes at minimum spanning tree structure. To do this we introduce m-minimum fuzzy triangular interval based estimation techniques. Fuzzy credibility distribution function with membership value m through [10] is used to generate the interval based cost between two nodes (mobile device) in graphical MANET structure.

I.J. Computer Network and Information Security, 2018, 6, 16-26

Improving the Performance of Fuzzy Minimum Spanning Tree based Routing Process through P-Node Fuzzy Multicasting Approach in MANET

As we emphasis on real-life uncertain environment, so energy utilization value is also represented with interval estimation techniques with minimum and maximum energy limits. Here Heinzelman’s energy model [23] is used to represent residual energy in the interval based format for transferring and receiving data packets. Using fuzzy shortest path technique we generate different fuzzy interval based p-nodes fuzzy multicast minimum spanning trees(pFMMST),where p is less than or equal to the total number of nodes(n) present in the graphical structure. As these multicast groups have less number of nodes, so topological changes or link failures have fewer effects on interval-based routing path cost and energy usage as compared to all the nodes(n) at minimum spanning tree of MANET graphical structure. Because the generated route cost and energy both are interval based, so errors due to deviation in between upper and lower limit value often generate the imperfection in optimization. Here the possibility of such type of errors also can be reduced by introducing pFMMST in place of normal FMST approach. Considering all these possible real lives constrains as uncertainty we try to improve the FMST based routing path cost, energy utilization and reduce the interval based data deviation related errors for MANET through the optimized method of the mminimum fuzzy triangular interval based value of pFMMST. To analyze the overall details we organize the rest of the paper in the following way: In section 2, the mathematical models are generated to simulate our dataset. In Section 3, we explain the fuzzy shortest path approach in p-node fuzzy multicasting minimum spanning tree and discuss the improvement of routing cost, energy utilization of nodes and possibility of errors due to interval based upper and lower limit data deviation, at FMST through pFMMST approach. Through Section 4, an example MANET network structure is taken to generate simulation techniques using fuzzy mathematical modeling. Where different tables and graphical representations are produced to show the improvement of routing cost, energy utilization and probable errors for interval-based route cost and energy deviation in pFMMST as compare to FMST. Finally, at section 5, we conclude and generate an idea for further future improvement.

Here a network structure is considered, with n nodes, which communicate amongst them using the wireless link. Considering our assumption if we take n node minimum spanning tree and any packet is forwarded by a node, it should be accepted by all its neighbors. It is believed that while configuring a multicast network, the source node will send the packets to its multicast group members only where the other neighbor nodes, which are not the member of that multicast group, will not receive a single packet. Which supports the packets to move less routing path cost and the packet transferring energy utilization will also reduce. Here the concepts of m-minimum Copyright © 2018 MECS

triangular fuzzy variables and credibility distributed fuzzy variables are introduced. These are used for the route cost and energy utilization to transmit packets between two neighbor nodes. Few significant definitions, related to our discussion, as follows. A. Triangular Fuzzy number A Triangular fuzzy number 𝐴̃ = {𝑎1 , 𝑎2 , 𝑎3 } is interpreted with the membership functions and holds the below following conditions. 1. 2. 3.

From 𝑎1 to 𝑎2 the membership value gets increased. From 𝑎2 to 𝑎3 the membership value gets decreased. 𝑎1 ≤ 𝑎2 ≤ 𝑎3

Fig.1. Triangular Fuzzy Number

Suppose for a value x, the membership function is µ𝐴̃ (𝑥) , where 0 ≤ µ𝐴̃ (𝑥) ≤ 1 According to Fig.1 it can be defined as 0

𝑥−𝑎1

𝑎2 −𝑎1 𝑎2 −𝑥

µ𝐴̃ (𝑥) = {

𝑎3 −𝑎2

1

𝑖𝑓 𝑥 < 𝑎1 𝑖𝑓 𝑎1 ≤ 𝑥 ≤ 𝑎2 (1)

𝑖𝑓 𝑎2 < 𝑥 ≤ 𝑎3 𝑖𝑓 𝑥 > 𝑎3

}

B. Credibility Distribution of Fuzzy Variable The credibility distribution β: ψ→ [0, 1] of a fuzzy variable 𝛷 is defined as (Zhou et al., 2015) 𝛽(𝑥) = 𝐶𝑟{𝜆 ∈ 𝛳| 𝛷(𝜆) ≤ 𝑥}

(2)

Where Cr{ Φ ≤ r} =

II. MATHEMATICAL MODELING

17

1 2

(supx≤r µ(x) + 1 − supx>r µ(x))

(3)

Here µ is membership value and r is a real value. Where 0 ≤ µ ≤ 1 Applying equation (2) and equation (3) at equation (1) we get β(x) = Cr{ Φ ≤ x} = 0 if x < a1 x−a1 if a1 < x ≤ a2

2(a2 −a1 ) x+a3 −2a2

{

2(a3 −a2 )

(4)

if a2 < x ≤ a3

1 if x > a3

}

I.J. Computer Network and Information Security, 2018, 6, 16-26

18

Improving the Performance of Fuzzy Minimum Spanning Tree based Routing Process through P-Node Fuzzy Multicasting Approach in MANET

From this credibility distribution, we get inverse credibility distribution values, which are basically interval, estimated fuzzy values with lower and upper bound limit. 𝛽−1 (𝑚) = 𝑎1 + 2(𝑎2 − 𝑎1 )𝑚 𝑖𝑓 0 ≤ 𝑚 ≤ 0.5 { } 2𝑎2 − 𝑎3 + 2(𝑎3 − 𝑎2 )𝑚 𝑖𝑓 0.5 < 𝑚 ≤ 1

(5)

Which is m-minimum interval based fuzzy data with lower and upper limit. Now using Mean-max technique we can defuzzify all the interval values into crisp data and get the best possible path cost of total n nodes. 𝑎1𝑖 +2(𝑎2𝑖 −𝑎1𝑖 )𝑚+2𝑎2𝑖 −𝑎3𝑖 +2(𝑎3𝑖 −𝑎2𝑖 )𝑚

𝑤 ̃𝑖 = [

2

]

(8)

C. Fuzzy Weighted Graph

D. Fuzzy Residual Energy

A normal weighted graph is a collection of vertices and edges with specific weights.

Here we have used Heinzelman’s energy model for transmitting packets from one router to another router. While transferring packets in between two nodes, at the sender side, the energy is required to transmit the packet and at receiver side energy is required to receive the packet. Suppose, k bit data is transmitted and distance between two nodes is d (Here distance unit is considered as meter), according to Heinzelman’s energy model Energy consumption for transmitting packet is given as

So,𝐺 = (𝑉, 𝐸, 𝑊) Where 𝑉 = {𝑣2 , 𝑣2 , … , 𝑣𝑛 } E= {𝑒1 , 𝑒2 , … , 𝑒𝑘 } , 𝑊 = {𝑤1 , 𝑤2 , … , 𝑤𝑘 } Here total number of vertices, edges and weights are n, k, k respectively. 𝑛(𝑛−1) If 𝑘 = , t he graph will be complete weighted 2 graph. In a fuzzy weighted graph, the cost or weight of the edges between two nodes will be fuzzy. Here, we consider m-minimum triangular fuzzy data to represent the cost. So, for fuzzy weighted graph

𝐸𝑇 (𝑘, 𝑑) = 𝐸𝑒𝑙𝑒𝑐 . 𝑘 + 𝐸𝑎𝑚𝑝 . 𝑘. 𝑑 2

(9)

Energy consumption for receiving packet is given as 𝐸𝑅 (𝑘) = 𝐸𝑒𝑙𝑒𝑐 . 𝑘

(10)

Where 𝐸𝑒𝑙𝑒𝑐 = 50𝑛𝐽/𝑏𝑖𝑡, 𝐸𝑎𝑚𝑝 =

̃) 𝐺̃ = (𝑉, 𝐸, 𝑊

100𝑝𝐽 𝑏𝑖𝑡

/𝑚2

Now for interval based route cost from equation (7) 1 Where,

d̃i = [a1i + 2(ai2 − a1i )m, 2ai2 − ai3 + 2(ai3 − ai2 )m]; 0≤m ≤1

𝑉 = {𝑣2 , 𝑣2 , … , 𝑣𝑛 } ̃ = {𝑤 E= {𝑒1 , 𝑒2 , … , 𝑒𝑘 } , 𝑊 ̃1 , 𝑤 ̃2 , … , 𝑤 ̃𝑘 } 𝑤 ̃𝑖 = {𝑎1𝑖 , 𝑎2𝑖 , 𝑎3𝑖 }

𝑑̃𝑖 = [𝑑1𝑖 , 𝑑2𝑖 ]

This is basically a triangular fuzzy number. It means, here we take the cost or weight of the edge 𝑒𝑖 as fuzzy data. Using credibility distribution equation (4) on 𝑤 ̃𝑖 we get

2(𝑎2𝑖 −𝑎1𝑖 ) 𝑥+𝑎3𝑖 −2𝑎2𝑖 2(𝑎3𝑖 −𝑎2𝑖 )

{

𝑖𝑓 𝑎1𝑖 < 𝑥 ≤ 𝑎2𝑖 (6)

𝑖𝑓 𝑎2𝑖 < 𝑥 ≤ 𝑎3𝑖

1 𝑖𝑓 𝑥 > 𝑎3𝑖

𝐸̃𝑇𝑖 (𝑘, 𝑑̃ 𝑖 ) = [𝐸̃𝑇𝑖 (𝑘, 𝑑̃1𝑖 ), 𝐸̃𝑇𝑖 (𝑘, 𝑑̃2𝑖 )]

(12)

2 𝐸̃𝑇𝑖 (𝑘, 𝑑̃1𝑖 ) = 𝐸𝑒𝑙𝑒𝑐 . 𝑘 + 𝐸𝑎𝑚𝑝 . 𝑘. 𝑑̃1𝑖 2 𝐸̃𝑇𝑖 (𝑘, 𝑑̃2𝑖 ) = 𝐸𝑒𝑙𝑒𝑐 . 𝑘 + 𝐸𝑎𝑚𝑝 . 𝑘. 𝑑̃2𝑖

Fuzzy receiving energy }

𝐸̃𝑅𝑖 (𝑘) = 𝐸𝑒𝑙𝑒𝑐 . 𝑘

Now applying inverse credibility distribution equation (5), we can get the cost 𝑤 ̃1 of edge 𝑒𝑖 as interval estimated manner. Where 𝑤 ̃𝑖 = [𝑎1𝑖 + 2(𝑎2𝑖 − 𝑎1𝑖 )𝑚, 2𝑎2𝑖 − 𝑎3𝑖 + 2(𝑎3𝑖 − 𝑎2𝑖 )𝑚]; 0≤𝑚 ≤1 (7) Copyright © 2018 MECS

So, fuzzy transmission energy,

Where

𝛽𝑖 (𝑥) = 𝐶𝑟{ 𝛷𝑖 ≤ 𝑥} = 0 𝑖𝑓 𝑥 ≤ 𝑎1𝑖 𝑥−𝑎1𝑖

(11)

(13)

Total residual energy consumption for transmitting one packet is 𝐿 𝐻 𝐸̃𝑅𝑖 = [𝐸̃𝑅𝑖 , 𝐸̃𝑅𝑖 ]

(14)

I.J. Computer Network and Information Security, 2018, 6, 16-26

Improving the Performance of Fuzzy Minimum Spanning Tree based Routing Process through P-Node Fuzzy Multicasting Approach in MANET

𝐸̃𝑅𝑖 = [𝐸̃𝑇𝑖 (𝑘, 𝑑̃1𝑖 ) + 𝐸̃𝑅𝑖 (𝑘), 𝐸̃𝑇𝑖 (𝑘, 𝑑̃2𝑖 ) + 𝐸̃𝑅𝑖 (𝑘)]

Here according to equation (7) 𝑤 ̃𝑖 = [𝑎1𝑖 + 2(𝑎2𝑖 − 𝑎1𝑖 )𝑚, 2𝑎2𝑖 − 𝑎3𝑖 + 2(𝑎3𝑖 − 𝑎2𝑖 )𝑚]; 0 ≤ 𝑚 ≤ 1

Where 𝐿 𝐸̃𝑅𝑖 = 𝐸̃𝑇𝑖 (𝑘, 𝑑̃1𝑖 ) + 𝐸̃𝑅𝑖 (𝑘) ̃𝐸𝑅𝑖𝐻 = 𝐸̃𝑇𝑖 (𝑘, 𝑑̃2𝑖 ) + 𝐸̃𝑅𝑖 (𝑘)

After defuzzification ̃𝑖

𝐿

19

𝐻

̃𝑖

𝑖 𝐸 +𝐸 𝐸̃𝑅𝑐 = [ 𝑅 𝑅 ]

(15)

2

Now using Mean-max technique we can defuzzify all the interval values into crisp data and get the best possible weight. ̃ ), a spanning tree 𝑇 0 is For a given graph𝐺̃ = (𝑉, 𝐸, 𝑊 called FMST When, 𝑊(𝑇 0 , 𝑤 ̃) ≤ 𝑊(𝑇, 𝑤 ̃)

(19)

E. Deviation of Upper and Lower Limit Fuzzy Value Where

For 𝑒𝑖 edge the route cost is 𝑤 ̃𝑖 from equation (7) and (8) we get 𝑤 ̃𝑖 = [𝑎1𝑖 + 2(𝑎2𝑖 − 𝑎1𝑖 )𝑚, 2𝑎2𝑖 − 𝑎3𝑖 + 2(𝑎3𝑖 − 𝑎2𝑖 )𝑚]; 0 ≤ 𝑚 ≤ 1 =[𝑤 ̃𝑖1 , 𝑤 ̃𝑖2 ]

G. Fuzzy Dijkstra Shortest Path

So, the deviation for lower and upper limit route cost is 2

2

̃ 𝑖1 −𝑤𝑖𝑐 ) +(𝑤 ̃ 𝑖2 −𝑤𝑖𝑐 ) (𝑤

𝑠𝑑(𝑤 ̃𝑖 ) = √[

2

]

(16)

𝑤 ̃𝑖1 + 𝑤 ̃𝑖2 𝑤𝑖𝑐 = [ ] 2 Using equation (14) and (15) we get deviation for lower and upper limit of residual energy 𝐿

= √[

𝑖 2

𝐻

𝑖 2

𝑖 −𝐸 𝑐 ) +(𝐸 𝑖 −𝐸 𝑐 ) ̃𝑅 ̃𝑅 ̃𝑅 (𝐸̃𝑅

2

]

(17)

F. Fuzzy Minimum Spanning Tree Considering fuzzy weighted graph we know, ̃) 𝐺̃ = (𝑉, 𝐸, 𝑊 Where 𝑉 = {𝑣2 , 𝑣2 , … , 𝑣𝑛 } ̃ = {𝑤 {𝑒 E= 1 , 𝑒2 , … , 𝑒𝑘 } , 𝑊 ̃1 , 𝑤 ̃2 , … , 𝑤 ̃𝑘 } 𝑤 ̃𝑖 = {𝑎1𝑖 , 𝑎2𝑖 , 𝑎3𝑖 }

𝑊(𝑇, 𝑤 ̃) = ∑𝑒𝑖 ∈𝑇 𝑤 ̃𝑖

Dijkstra_shortestpath (v, cost, distance, n) // for 1  i  n distance[i] represents the shortest path distance from node v to i { for j: =1 to n do { //initialize S. S[v]:=false; distance[j]:=cost [v,j]; } S[v]:=true; distance[v]:=0.0; //insert v in S. for m: =2 to n do { // determine n-1 path from v. // choose u from those sets of nodes which are unavailable at S and distance[u] will be minimum. S[u]:=true; // put u in S. for (each p adjacent to u with S[p] = false) do //Update distances. if (distance[p]>distance[u] + cost [u, p])) then distance[p]:=distance[u] + cost [u, p]; } } 2. Fuzzy Dijkstra shortest path algorithm

̃ ) is A fuzzy spanning tree 𝑇 = (𝑉, 𝑆̃ ) of 𝐺̃ = (𝑉, 𝐸, 𝑊 a connected acyclic sub graph, containing all vertices, where S is the set of edges, contained in T. So, 𝑆̃ ⊆ 𝐸 The weight of a spanning tree T can be formulated as

Copyright © 2018 MECS

Famous Dutch scientist Edsger Dijkstra established the Dijkstra algorithm in the year of 1956 and officially published in 1959. Here Dijkstra’s algorithm is used generate the shortest route between any two nodes (Digital device) in MANET network structure. 1. Conventional Dijkstra shortest path algorithm

Where

𝑠𝑑(𝐸𝑅𝑖 )

W(T 0 , w ̃ ) = [W(T 0 , w ̃ L ), W(T 0 , w ̃ H )]

(18)

As we are working on fuzzy environment, according to conventional Dijkstra algorithm usable crisp point estimated path cost is changed to interval based path cost. For processing Dijkstra algorithm we use the following steps. 1.

At first, we use inverse credibility on m-minimum

I.J. Computer Network and Information Security, 2018, 6, 16-26

20

2. 3.

Improving the Performance of Fuzzy Minimum Spanning Tree based Routing Process through P-Node Fuzzy Multicasting Approach in MANET

triangular fuzzy path cost of the network and generate all the path cost as interval based fuzzy number. Here each cost of network path has two values. One is upper bound and another is lower bound. Using the upper bound value and lower bound value we generate the shortest path for each pair of nodes.

Suppose using Dijkstra algorithm on upper bound and lower bound values, we get the shortest paths of (𝑣𝑖 , 𝑣𝑗 ) are 𝑙1 and 𝑙2 respectively. 4.

Now the shortest path of (𝑣𝑖 , 𝑣𝑗 ) is considered as [𝑙1 , 𝑙2 ] Using the middle of maxima method according to equation (8), we can defuzzify this interval based value as 𝑙 , where 𝑙 = 𝑙1 + 𝑙2 ⁄2 is minimum shortest path cost between(𝑣𝑖 , 𝑣𝑗 )

III. P-NODE FUZZY MULTICASTING MINIMUM SPANNING TREE ̃) For fuzzy weighted graph 𝐺̃ = (𝑉, 𝐸, 𝑊 Where,

Statement 1. If for any MANET network total number of active nodes(digital device) are n to route the packets, the total number of multicasting groups(act as separate network structure for transferring packet) with two or more active nodes will be, 2n − (1 + n) Explanation: According to the set theory we can say, Cardinality of 𝑉 = 𝑛 So, the power set for V is 𝑃|𝑉|, where all possible subsets of nodes are included. Now, 𝑃|𝑉| = 2𝑛 where there is a null subset and n number of single node subsets. For creating multicasting group we can omit (1 + 𝑛) number of subsets. So, the number of subsets of nodes with at least 2 or more nodes will be 2𝑛 − (1 + 𝑛). Here we are interested to create multicast group MANET, each with at least 3 nodes. So number of multicast groups will be 2𝑛 − (1 + 𝑛 + 𝑛𝐶2 ). Statement 2. The network routing path cost of pFMMST is less than or equal to FMST of G for a particular source and destination node. Explanation: As, 𝑝 ≤ 𝑛𝑆 from equation (19) we get ∑pi=1 w ̃ i ≤ WnS (T 0 , w ̃)

(22)

𝑉 = {𝑣2 , 𝑣2 , … , 𝑣𝑛 } ̃ = {𝑤 E={𝑒1 , 𝑒2 , … , 𝑒𝑘 } , 𝑊 ̃1 , 𝑤 ̃2 , … , 𝑤 ̃𝑘 } 𝑤 ̃𝑖 = {𝑎1𝑖 , 𝑎2𝑖 , 𝑎3𝑖 }

it means that, routing path cost of pFMMST is less than FMST. For other pFMMSTs’ also equation (15) will get satisfied.

This is basically triangular fuzzy number. It means, here we take the cost or weight of the edge 𝑒𝑖 as fuzzy data. Using fuzzy Dijkstra shortest path technique, we can generate all pair shortest paths. Suppose,

Statement 3. The network residual energy for routing the packets through pFMMST is less than or equal to the network residual energy of the FMST of G Explanation: Total fuzzy residual energy utilization in FMST can be calculated from equation (14) and (15) as 𝐿

Where, the number of nodes is p. Where 𝑆𝑝𝑎𝑡ℎ(𝑣𝑖 , 𝑣𝑗 ) , signifies the shortest route between two nodes (𝑣𝑖 , 𝑣𝑗 ) and 𝑝 ≤ 𝑛𝑠 𝑛𝑠 = number of nodes in FMST for a particular source and destination. So, the total path cost is. 𝑝 = [∑𝑙=1 𝑎1𝑙 + 2(𝑎2𝑙 2(𝑎3𝑙 − 𝑎2𝑙 )𝑚]

∑𝑃𝑙=1 𝑤 ̃𝑖

−

𝑝 𝑎1𝑙 )𝑚 , ∑𝑖=1 2𝑎2𝑙

−

𝑎3𝑙

+ (20)

Now using Mean-max technique we can defuzzify all the interval values into crisp data and get the best possible cost of total p nodes 𝑝

𝑝

∑𝑙=1 𝑎1𝑙 +2(𝑎2𝑙 −𝑎1𝑙 )𝑚+∑𝑙=1 2𝑎2𝑙 −𝑎3𝑙 +2(𝑎3𝑙 −𝑎2𝑙 )𝑚

∑𝑃𝑙=1 𝑤 ̃𝑖 = [

2

] (21)

Where 𝑣𝑖 → 𝑣𝑟 , 𝑣𝑟 → 𝑣𝑞 up to 𝑣𝑠 → 𝑣𝑗 the minimum weights are {𝑤 ̃1 , 𝑤 ̃2 , … , 𝑤 ̃𝑝 } respectively.

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𝐻

𝑛𝑆 ̃ 𝑖 𝑛𝑆 ̃ 𝑖 𝑆 ̃𝑖 ∑𝑛𝑖=1 𝐸𝑅 = [∑𝑖=1 𝐸𝑅 , ∑𝑖=1 𝐸𝑅 ]

𝑆𝑝𝑎𝑡ℎ(𝑣𝑖 , 𝑣𝑗 ) 𝑣𝑖 → 𝑣𝑟 → 𝑣𝑞 … . 𝑣𝑠 → 𝑣𝑗

(23)

Again using the same procedure and equations (14), (15) the total residual energy for pFMMSTs will be 𝐿

𝐻

∑𝑝𝑙=1 𝐸̃𝑅𝑙 = [∑𝑝𝑙=1 𝐸̃𝑅𝑙 , ∑𝑝𝑙=1 𝐸̃𝑅𝑙 ]

(24)

As, 𝑝 ≤ 𝑛𝑠 𝑛𝑆 ̃ 𝑖 ∑𝑝𝑙=1 𝐸̃𝑅𝑙 ≤ ∑𝑖=1 𝐸𝑅

(25)

Which signifies that, total fuzzy residual energy utilization in pFMMST is less than or equal to FMST. Comparison of p-node multicast minimum spanning tree and minimum spanning tree is one of the significant areas to increase the performance of MANETs. But in real life uncertain environment, it is found that point estimated result analysis in the crisp environment does not generate satisfactory results because different physical constrains in MANET structure create the variation, which does not generate any optimum decision oriented result. So the interval estimated fuzzy system is applied to improve the quality of decision making.

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Improving the Performance of Fuzzy Minimum Spanning Tree based Routing Process through P-Node Fuzzy Multicasting Approach in MANET

Statement 4. The deviation of upper and lower limit value in interval based total route cost in pFMMST is less or equal to FMST. Explanation: 𝑝 𝑝 ∑𝑝𝑙=1 𝑤 ̃𝑙 = [∑𝑙=1 𝑤 ̃𝑙𝐿 , ∑𝑙=1 𝑤 ̃𝑙𝐻 ]

According to equation (16) 𝑝

𝑠𝑑(∑𝑙=1 𝑤 ̃𝑙 ) = 𝑝

2

𝑝

𝑝

𝑝

̃ 𝑙𝐿 −∑𝑙=1 𝑤𝑙𝑐 ) +(∑𝑙=1 𝑤 ̃ 𝑙𝐻 −∑𝑙=1 𝑤𝑙𝑐 ) (∑𝑙=1 𝑤

√[

2

2

]

(26)

From equation (19) 𝑠𝑑(𝑊(𝑇 0 , 𝑤 ̃)) = √(𝑊(𝑇

2 2 0 ,𝑤 ̃ 𝐿 )−𝑊(𝑇 0 ,𝑤 ̃ 𝑐 )) +(𝑊(𝑇 0 ,𝑤 ̃ 𝐻 )−𝑊(𝑇 0 ,𝑤 ̃ 𝑐 ))

2

𝐿

𝑝

2

𝑝

𝑝

𝐻

21

𝑝

(∑𝑙=1 𝐸̃𝑅𝑙 − ∑𝑙=1 𝐸̃𝑅𝑐 ) + (∑𝑙=1 𝐸̃𝑅𝑙 − ∑𝑙=1 𝐸̃𝑅𝑐 ) 𝑛

𝐿

𝑛

2

𝑛

𝐻

𝑛

2

2

𝑆 ̃𝑖 𝑆 ̃𝑐 𝑆 ̃𝑖 𝑆 ̃𝑐 ≥ (∑𝑖=1 𝐸𝑅 − ∑𝑖=1 𝐸𝑅 ) + (∑𝑖=1 𝐸𝑅 − ∑𝑖=1 𝐸𝑅 ) (32)

But the possibility of above condition (as per equation (32)), is exceptionally rare where the error rate for the imprecise residual energy of pFMMST is greater than FMST. According to the normal statistical convention, with the increase of standard deviation, the error is also increased. Here the interval based imprecise data is considered, where the deviation of upper and lower limit value should be justified to generate an optimized result. Here from equation (28) and (31), it is shown that pFMMST gives much more optimized deviation of interval based route cost and residual energy utilization(for most of the pair of nodes) than FMST. In it pFMMST gives lower or equal value standard deviation, compared to FMST, for route cost and energy usage.

(27) IV. SIMULATION STUDY AND RESULT ANALYSIS

Where 𝑊(𝑇 0 , 𝑤 ̃ 𝑐) = [

̃ 𝐿 )+𝑊(𝑇 0 ,𝑤 ̃ 𝐻) 𝑊(𝑇 0 ,𝑤 2

]

𝑝 As ∑𝑙=1 𝑤 ̃𝑙 ≤ 𝑊(𝑇 0 , 𝑤 ̃ 𝑐)

So from equation (24) and (25) it can be concluded 𝑝 (𝑇 0 , 𝑤 ̃) ≥ 𝑠𝑑(∑𝑙=1 𝑤 ̃𝑙 )

(28)

Statement 5. The deviation of upper and lower limit value in the interval based total residual energy for most of the pair of nodes in pFMMST, is less or equal to FMST. Explanation: According to equation (17), we get

In order to study the path distance and packet sending energy optimization in a MANET, we generate FMST and pFMMST from a fuzzy weighted graph. Here the path distance and residual energy are considered with meter and nano joule as unit respectively. We use Maple12, Math Type 6 and Microsoft Office Excel 2007 as simulators to design the fuzzy graph through which we calculate all pairs of the fuzzy minimum shortest path using Dijkstra algorithm and FMST, using Kruskal algorithm. Table 1. Weight Details of the Fuzzy Graph

𝑝

𝑠𝑑(∑𝑙=1 𝐸̃𝑅𝑙 ) = 𝑙𝐿

𝑝

𝑝

𝑐

2

√(∑𝑙=1 𝐸̃𝑅 −∑𝑙=1 𝐸̃𝑅)

𝑝

𝐻

𝑝

2

𝑙 −∑ ̃𝑐 +(∑𝑙=1 𝐸̃𝑅 𝑙=1 𝐸𝑅 ) 2

(29)

𝑛

𝑆 ̃𝑖 𝑠𝑑(∑𝑖=1 𝐸𝑅 ) = 𝑛𝑆

𝑖𝐿

𝑛𝑆

𝑐

2

√(∑𝑖=1 𝐸̃𝑅 −∑𝑖=1 𝐸̃𝑅)

𝐻

2

𝑛𝑆 𝑖 𝑛𝑆 𝑐 +(∑𝑖=1 𝐸̃𝑅 −∑𝑖=1 𝐸̃𝑅 )

2

(30)

𝑛𝑆 ̃ 𝑖 𝑝 As ∑𝑙=1 𝐸̃𝑅𝑙 ≤ ∑𝑖=1 𝐸𝑅 For 𝑝 ≤ 𝑛𝑠 so from equation (29), (30) it can be depicted that for most of the pair of nodes 𝑝

𝑛

𝑆 ̃𝑖 𝑠𝑑(∑𝑙=1 𝐸̃𝑅𝑙 ) ≤ 𝑠𝑑(∑𝑖=1 𝐸𝑅 )

(31)

On the other hand, for very few pair of nodes the following exception may occur:

Copyright © 2018 MECS

From Fig.2, we generate the FMST and calculate the interval based fuzzy weight 𝑊𝑛𝑆 (𝑇 0 , 𝑤 ̃) of the acyclic tree, given in Table 2.

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22

Improving the Performance of Fuzzy Minimum Spanning Tree based Routing Process through P-Node Fuzzy Multicasting Approach in MANET

Fig.2. Fuzzy Graph (MANET structure) Table 2. Weight Details of FMST

Fig.4. (p=5, 1-10) pFMMST (cost= [36.6, 38.4])

Fig.5. Using (n=11, 1-10) FMST (cost= [42.6, 47.4]) Table 3. Comparative Study of Route Cost of pFMMST and FMST

Fig.3. FMST (MANET structure)

As we know p=number of nodes in the multicast tree and 𝑛𝑆 =no of nodes in the minimum spanning tree. According to equation (18), (26), (27) we generate Table 3. Here Table 3 shows a brief comparative study of following areas. 1. 2.

Edge weight of pFMMST and FMST. The standard deviation of edge weight of pFMMST and FMST.

Copyright © 2018 MECS

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Improving the Performance of Fuzzy Minimum Spanning Tree based Routing Process through P-Node Fuzzy Multicasting Approach in MANET

23

path cost, the risk of environmental constrain, topological changes and noise inclusion are increased. Here for FMST from the route cost [42.6, 47.4], the standard deviation of upper and lower limit value is 2.4(using equation 25). In pFMMST for route cost [36.6, 38.4], the standard deviation of upper and lower limit value is 0.9(using equation 24). According to Table3 and Fig.8, it is clear, due to atmospheric change and physical constrains, the variation of route cost at the time of data transmission in pFMMST is less or equal to FMST. So, pFMMST has less possibility for error occurrence in route cost as compare to FMST.

Table 4. Residual Energy of Nodes in MANET

Table 5. Comparative Study of Residual Energy of pFMMST and FMST

Fig.6. Packet Transfer Path Cost in FMST in MANET

A. Comparison of edge cost between pFMMST and FMST As we know for n nodes MANET, total 2n -(1 + n + nC2 ) number of p node multicast groups can be created, where 𝑝 ≥ 3 .According to Fig. 4 and Fig. 5, it is observed that for transferring the packet from node 1 to 10 using FMST ,the overall interval based path cost is [42.6, 47.4], where for pFMMST, it will be [36.6, 38.4]. Here we can analyze the route cost more sensibly for both the cases with upper and lower limit interval based format, which signifies the variation of cost within a range due to the effect of real life uncertainty. In case of pFMMST, packet transferring path is 𝑣1 → 𝑣2 → 𝑣5 → 𝑣8 → 𝑣10 Using FMST, to reach same destination from same source, packet transferring path is 𝑣1 → 𝑣2 → 𝑣3 → 𝑣4 → 𝑣7 → 𝑣6 → 𝑣5 → 𝑣8 → 𝑣9 → 𝑣11 → 𝑣10 Involvement of a large number of nodes has the maximum possibility for data loss in MANET. Because due to increasing the node numbers, the possibility of Copyright © 2018 MECS

Fig.7. Packet Transfer Path Cost in pFMMST in MANET

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24

Improving the Performance of Fuzzy Minimum Spanning Tree based Routing Process through P-Node Fuzzy Multicasting Approach in MANET

Fig.9. Fuzzy Residual Energy for FMST in MANET

Fig.8. Edge Cost Standard Deviation of FMST and pFMMST

So, here interval based route costs in fuzzy upper and lower limit format practically justifies that pFMMST is more improved minimum spanning tree method with less data loss to transfer the data packets in an uncertain environment for MANET. B. Comparison of residual energy between pFMMST and FMST For FMST, according to equation (23) and Fig.5, 𝑛𝑆 = 11 Where i = 1,2,3,4,7,6,5,8,9,11,10 and m=0.8 nS

∑

i=1

̃iR = [8201.44 , 8231.488] E

Fig.10. Fuzzy Residual Energy for pFMMST in MANET

For pFMMST, according to equation (24) and Fig.4, 𝑝=5 Where 𝑙 = 1,2,5,8,10 and m=0.8 p

∑

l=1

̃iR = [3570.72 , 3597.504] E

So, here it is shown that, for transferring packets from node 1 to node 10, total residual energy utilization at pFMMST is smaller than energy utilization at FMST. Considering (1, 10) node pair, for FMST from the energy utilization [8201.44, 8231.488], the standard deviation of upper and lower limit value is 15.023(using equation 30) In pFMMST for energy utilization [3570.72, 3597.504], the standard deviation of upper and lower limit value is 13.39(using equation 29). According to Table5 and Fig.11, it shows that for most of the pair of nodes, pFMMST has less possibility for error occurrence than FMST in energy utilization. Copyright © 2018 MECS

Fig.11. Residual Energy Standard Deviation of FMST and pFMMST

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Improving the Performance of Fuzzy Minimum Spanning Tree based Routing Process through P-Node Fuzzy Multicasting Approach in MANET

V. CONCLUSIONS Here we try to implement m-minimum fuzzy triangular number in interval based format, using inverse credibility distribution to represent the minimum spanning tree and pFMMST of MANET. Where we compare the fuzzy path cost, total fuzzy residual energy and interval-based data deviation error for transferring the packets through both the acyclic tree structure of MANET and generate a conclusive decision. Here the reason for applying fuzzy concept is to attach the real-life constrains like rapid topological changes, insertion of noise or climatic variation with the architecture and achieve the most optimized result in MANET. Using the interval based fuzzy concept, it makes easier to generate approximated values, in place of contradictory point estimated results. Here in MANET structure we only consider the digital devices as nodes, but we did not consider separate sensor device as an active node. In future we can try to apply wireless sensor architecture with MANET in a fuzzy environment, using the artificial neural network for training the network path to generate more optimized results.

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Authors’ Profiles Soham Bandyopadhyay is a Lecturer in the Dept. of Computer Science and Technology at Dr.B.C. Roy Polytechnic, Durgapur, India. He received B.TECH in Computer science and Engg. from West Bengal University of Technology, MBAfrom SMU, M.TECH from National Institute of Technology, Durgapur India.His research interests are in the areas of wireless ad-hoc network systems, soft computing, fuzzy logic.

Sunil Karforma is a Professor in the Dept.of Computer Science at The University of Burdwan, West Bengal, India. He received B.TECH and M.TECH in Computer Science and Engineering from Jadavpur University, West Bengal, India. He received his Ph.D from The University of Burdwan, West Bengal, India in Computer Science. His research interests are in the areas of wireless network systems, e-commerce, network security, soft computing.

How to cite this paper: Soham Bandyopadhyay, Sunil Karforma,"Improving the Performance of Fuzzy Minimum Spanning Tree based Routing Process through P-Node Fuzzy Multicasting Approach in MANET", International Journal of Computer Network and Information Security(IJCNIS), Vol.10, No.6, pp.16-26, 2018.DOI: 10.5815/ijcnis.2018.06.02

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